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In this paper, a new identity-based signature (HIBS) scheme is proposed. Our new scheme has some advantages over the available schemes: the private keys size shrinks as the identity depth increases and the signature size is a constant as it consists of three group elements. Furthermore, under the selective-identity security model, we reduce its security to the l-CDH problem. This assumption is more natural than many of the hardness assumptions recently introduced to HIBS in the standard model.